6.1 Motivation

We have seen that generic convex problems, under mild computability and boundedness assumptions, are polynomially solvable, e.g., by the ellipsoid method. This result is extremely important theoretically; however, from the practical viewpoint it is essentially no more than an existence theorem. Indeed, the complexity bounds for the ellipsoid method, although polynomial, are not very attractive. By Theorem 5.2.1, when solving problem (5.2.9) with n design variables, the price of an accuracy digit (the cost of reducing the current inaccuracy ? by factor 2) is O( n ²) calls to the first order and the separation oracles, plus O( n ⁴) arithmetic operations to process the answers of the oracles. Thus, even for problems with very simple objectives and feasible sets, the arithmetic price of an accuracy digit is O( n ⁴). Imagine what it takes then to solve a problem with, say, 1000 variables (which is still a small size for many applications). One could hope, of course, that the efficiency estimate stated in Theorem 5.2.1 is no more than a worst-case theoretical upper bound, while the actual behavior of the ellipsoid method is typically much better than the bound says. A priori this is not entirely hopeless; for example, the LP simplex method is an extremely powerful computational tool, despite its disastrously bad (exponential in the size of an instance) worst-case efficiency estimate. Unfortunately, practice demonstrates that the ellipsoid method does work according to its theoretical efficiency estimate, and...